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Rigidity of closed minimal hypersurface in $\mathbb{S}^5$

Pengpeng Cheng, Tongzhu Li

Abstract

Let $M^4\to \mathbb{S}^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ in a $5$-dimensional sphere $\mathbb{S}^5$. In this paper, we prove that if $3$-mean curvature $H_3$ and the number $g$ of the distinct principal curvatures are constant, then $M^4$ is an isoparametric hypersurface, and the value of $S$ can only be $0, 4, 12$. This result supports Chern Conjecture.

Rigidity of closed minimal hypersurface in $\mathbb{S}^5$

Abstract

Let be a closed immersed minimal hypersurface with constant squared length of the second fundamental form in a -dimensional sphere . In this paper, we prove that if -mean curvature and the number of the distinct principal curvatures are constant, then is an isoparametric hypersurface, and the value of can only be . This result supports Chern Conjecture.

Paper Structure

This paper contains 3 sections, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1']}

Theorems & Definitions (1)

  • proof